pwr4exp
Power Analysis for Research Experiments

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pwr4exp: Power Analysis for Experimental Designs

pwr4exp: Power Analysis for Experimental Designs
In pwr4exp: Power Analysis for Research Experiments 

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(pwr4exp)

Introduction

pwr4exp is an R package for performing statistical power analysis for experiments analyzed using linear mixed models (LMM). It supports power calculations for omnibus F-tests (ANOVA-like tests) and specific contrasts (t-tests) for Gaussian response variables, employing the Satterthwaite approximation for degrees of freedom.

Key features:

The workflow involves two main steps:

  1. Creating a design object: Defining the experimental structure and specifying parameters (fixed effects, (co-)variances).

  2. Calculating power: Computing the power of F-tests or t-tests for the effects of interest.

Creating a design object {#design}

A design object in pwr4exp is a list containing design matrices, fixed effects, variance-covariance components, and other higher-level parameters created by the design-generating functions. It is not an experimental design in the traditional sense, where randomization and unit allocation are performed, but rather a container for all the information necessary to conduct power analysis.

Two approaches are provided to define a design:

The mkdesign function {#mkdesign}

The function mkdesign is versatile for designs as long as the data frame is provided.

Usage

mkdesign(
  formula, data,
  beta = NULL, means = NULL,
  vcomp = NULL, sigma2 = NULL,
  correlation = NULL,
  template = FALSE,
  REML = TRUE
)

Arguments

Specific design functions {#standard_design}

While mkdesign provides full flexibility, pwr4exp also offers specific functions for common experimental designs. These functions allow users to define a design based on treatment structure and replications without manually creating a data frame.

Completely randomized design (CRD)

designCRD(treatments, label, replicates, formula, beta, means, sigma2, template = FALSE)

Randomized complete block design (RCBD)

designRCBD(treatments, label, blocks, formula, beta, means, vcomp, sigma2, template = FALSE)

Latin square design (LSD)

designLSD(treatments, label, squares, reuse, formula, beta, means, vcomp, sigma2, template = FALSE)

Crossover design

This is a special case of LSD where time periods and individuals act as blocks. Period blocks are reused when replicating squares.

designCOD(treatments, label, squares, formula, beta, means, vcomp, sigma2, template = FALSE)

Split-plot design (SPD)

designSPD(trt.main, trt.sub, label, replicates, formula, beta, means, vcomp, sigma2, template = FALSE)

The inputs for these functions are similar to mkdesign, except that data is replaced by predefined design characteristics such as treatment levels and replications.

Key inputs

  • treatments: A numeric vector specifying the number of levels for treatment factors. Multiple factors are arranged factorially. For example, treatments = c(3, 2) specifies two treatment factors one with 3 levels and another with 2 levels, forming a factorial treatment design.

  • trt.main and trt.sub: Define main-plot and subplot treatments for SPD, following the same rule for treatment.

  • label: (optional) A list assigning labels to factor levels. If not specified, default names are assigned. For one treatment factor, the default is list(trt = c("1", "2", ...)). For two factors, the default is list(facA = c("1", "2", ...), facB = c("1", "2", ...)), where "facA" and "facB" represent the two factors, and "1", "2", etc., represent the levels of each factor.

  • replicates: Number of experimental units per treatment in CRD or number of main plots (i.e., the number of experimental units per main-plot treatment) in SPD.

  • blocks: Number of blocks in RCBD.

  • squares: Number of squares in LSD or COD.

  • reuse: Specifies how squares are replicated in LSD. One of:

  • "row": reuse row blocks
  • "col": reuse column blocks
  • "both": reuse both row and column blocks
  • template: When set to TRUE, templates for beta, means, and vcomp are generated to indicate the required input order.

Each of these design-generating functions has a default formula based on the treatment structure (e.g., one factor or factorial factors). If formula is not specified, a default formula with main effects and all interactions (if applicable) is used internally. In RCBD, LSD, COD, and SPD designs, all block factors are fitted as random effects. The formula component in the output design object (a list) can be inspected.

Parameter templates {#template}

Templates help ensure the correct ordering of fixed effects (beta or means) and random effects (vcomp).

Below, an exemplary data frame is created to illustrate these templates. Note that this example does NOT represent a realistic design. The data frame includes:

  • Four categorical variables (fA, fB, fC, fD),
  • Two numerical variables (x, z).
 df1 <- expand.grid(
   fA = factor(1:2), # factor A with 2 levels
   fB = factor(1:2), # factor B with 2 levels
   fC = factor(1:3), # factor C with 3 levels
   fD = factor(1:3), # factor D with 3 levels
   subject = factor(1:10)  # 10 subjects
 )
 df1$x <- rnorm(nrow(df1))  # Numerical variable x
 df1$z <- rnorm(nrow(df1))  # Numerical variable z

beta Template {#fixeff_template}

The beta template displays the order of model coefficients in the fitted model.

This is particularly useful when specifying expected model coefficients directly as effect size measures and when the model includes complex interactions. For example, the expected values of the model coefficients for ~ fA*fB + x should be provided in the following sequence:

mkdesign( ~ fA * fB + x, df1)$fixeff$beta

means Template

For treatment factors, it may be more more convenient to provide means instead of beta. The means template represents either marginal or cell means for factors, depending on the presence of interactions. For numerical variables included in the model, regression coefficients remain necessary alongside alongside the means for factors.

Factors without interactions

If no interactions are present, marginal means for each factor level are required.

For example, for the model ~ fA + fB, means should be reported in the order specified by the template:

mkdesign(~ fA + fB, df1, template = TRUE)$fixeff$means

Interactions among factors

When factors interact, conditional (cell) means must be provided for all possible combinations of their levels.

For example, for a model with three interacting factors (fA, fB, fC), cell means are required for all 12 level combinations in the following order:

mkdesign(~ fA * fB * fC, df1)$fixeff$means

Numerical variables

For numerical variables, regression coefficients are required. If the model only includes numerical variables, the intercept must also be included. In this case, the values in means are identical to beta:

mkdesign(~ x + z, df1)$fixeff$means

When numerical variables interact, regression coefficients for both main effects and interaction terms must be included.

mkdesign(~ x * z, df1)$fixeff$means

Factor-by-numerical interactions

For interactions between factor and numerical variables, both Marginal means for the factor, and Regression coefficients for the numerical variable at each level of the factor must be provided.

For instance, in the model ~ fA * x, the means for levels of fA and the regression coefficients for x at each level of fA must be provided:

mkdesign(~ fA * x, df1)$fixeff$means

Combining Multiple Situations

The rules outlined for factor and numerical variables apply when multiple types of variables and interactions appear in the same model.

For example, consider a model including interactions among three factors (fA, fB, fC), factor-by-numerical interaction (fD * x), and main effects for another numerical variable (z).

mkdesign(~ fA * fB * fC + fD * x + z, df1)$fixeff$means

The required elements and their order in means are:

1. Means for each level of fD (positions 1–3)

2. Regression coefficient for z (position 4)

3. Regression coefficients for x at each level of fD (positions 5–7)

4. Cell means for all combinations of levels of fA, fB, and fC (positions 8–19)

vcomp Template {#vcomp_template}

The vcomp template represents variance-covariance matrices, where integer values indicate the order of variance components in the input vector.

For a single random effect, a template is unnecessary since it corresponds to a single variance value. For multiple random effects, the template specifies the sequence of variance and covariance components, ensuring proper alignment when specifying variance components in the model.

For example, consider a model that includes both a random intercept and a random slope for fA by subject:

mkdesign(~ fA * fB * fC * fD + (1 + fA | subject), df1)$varcov

The template specifies the following required inputs:

  1. Variance of the random intercept

  2. Covariance between the random intercept and slop

  3. Variance of the random slop (fA2)

Correlation structures

Although specifying complex variance-covariance structures, such as unstructured nlme::corSymm, may seem impractical for power analysis, various correlation structures from the nlme package can be applies.

The available correlation structures, as outlined in nlme::corClasses, including

Note: In nlme::corAR1() and nlme::corARMA() when p=1 and q=0, the time variable must be an integer. However, in pwr4exp, this restriction has been released, factors are also supported.

Power calculation {#power}

Once a design object has been created, calculating statistical power is straightforward.

Power Calculation for Omnibus F-Tests

The statistical power for F-tests (ANOVA-like tests) can be computed using the pwr.anova function.

Usage

pwr.anova(object, sig.level = 0.05, type = 3)

Arguments

Power Calculation for Specific Contrasts

To evaluate statistical power for t-tests on specific contrasts, use the pwr.contrast function.

Usage

pwr.contrast(object, which, by = NULL, 
  contrast = c("pairwise", "poly", "trt.vs.ctrl"),
  sig.level = 0.05, p.adj = FALSE, 
  alternative = c("two.sided", "one.sided"), strict = TRUE
)

Arguments

Power Calculation for Model Coefficients

The pwr.summary function computes the statistical power for testing model coefficients (t-tests).

Usage

pwr.summary(object, sig.level = 0.05)

Arguments

Practical Examples

Example 1. Completely Randomized Design

In this example, we create a CRD with one treatment factor.

Design parameters

  1. Treatments: 1 treatment factor with 4 levels.
  2. Replicates: 8 experimental units per treatment.
  3. Mean and effect size: Expected means for four levels: 35, 30, 37, 38
  4. Error variance: 15

Step 1: Creating the CRD

crd <- designCRD(
  treatments = 4,
  replicates = 8,
  means = c(35, 30, 37, 38),
  sigma2 = 15
)

Step 2: Power for omnibus test

To compute the power for the overall F-test (ANOVA):

pwr.anova(crd)

Step 3: Power for specific contrasts

a) Pairwise Comparisons

To compute power for pairwise differences:

pwr.contrast(crd, which =  "trt", contrast = "pairwise")

b) Polynomial Contrasts

To test for trends across treatment levels:

pwr.contrast(crd, which =  "trt", contrast = "poly")

c) Treatment vs. Control Comparison

The power for detecting differences between treatments and the control:

pwr.contrast(crd, which =  "trt", contrast = "trt.vs.ctrl")

d) Custom Contrast: All Treatments vs. Control

In addition to pre-defined contrast method, custom contrast vectors can be specified. For example, to compare the average of all treatments against the control, use the contrast vector c(-1, 1/3, 1/3, 1/3). The power for this test can be computed as:

pwr.contrast(crd, which =  "trt", contrast = list(trts.vs.ctrl = c(-1, 1/3, 1/3, 1/3)))

Adjusting for Multiple Comparisons

In real-world analysis, P-values often need adjustment for multiple comparisons (MCP). While most post-hoc methods cannot be applied directly in power analysis, we can lower the significance level to approximate MCP adjustments.

Using a More Conservative Significance Level

pwr.contrast(crd, which =  "trt", contrast = "pairwise", sig.level = 0.01)

Using the Bonferroni Correction

The pwr.contrast function includes an option for Bonferroni correction, though it may be overly conservative:

pwr.contrast(crd, which =  "trt", contrast = "pairwise", sig.level = 0.05, p.adj = TRUE)

Example 2. Randomized complete block design

In this example, we create a Randomized Complete Block Design (RCBD) with two treatment factors in a 2×2 factorial design.

Design Parameters

  1. Treatments: A 2x2 factorial design (factors: A and B).
  2. Replicates: 8 blocks.

  3. Expected Means: $$ \begin{array}{c|c|c} & B1 & B2 \ \hline A1 & 35 & 38 \ A2 & 40 & 41 \ \end{array} $$ Corresponding beta values are as follows (Optional):

  4. Intercept (A1B1): 35

  5. Effect of A2 alone: 5 units
  6. Effect of B2 alone: 3 units

  7. Interaction (A2:B2): -2 units (i.e., the combined effect of A2 and B2 is 2 units below the additive effects).

  8. Variance among blocks: 11.

  9. Error variance: 4.
  10. The total variance of the response variable (15) is decomposed into variance between blocks (11) and variance within blocks (4).

Step 1: Creating the RCBD

designRCBD(treatments = c(2, 2), blocks = 8, template = TRUE)

rcbd <- designRCBD(
  treatments = c(2, 2),
  blocks = 8,
  # beta = c(35, 5, 3, -2), # identical to means
  means = c(35, 40, 38, 41),
  vcomp = 11,
  sigma2 = 4
)

The default factor names and formula can be inspected in the design object:

unique(rcbd$deStruct$fxTrms$fixedfr)
rcbd$deStruct$formula

Treatment factors and factor levels can be renamed and relabeled, and the model formula can be changed. For example, if interactions are removed, marginal means for both factors must be provided instead of cell means.

designRCBD(treatments = c(2, 2), 
           label = list(factorA = c("A1", "A2"), factorB = c("B1", "B2")), 
           blocks = 8, 
           formula = ~ factorA + factorB + (1|block), 
           template = TRUE)

Step 2: Evaluate statistical power

To test overall effects of facA and facB

pwr.anova(rcbd)

To test differences between levels of facA, conditioned on facB:

pwr.contrast(rcbd, which = "facA", by = "facB")

Example 3. Latin square design

In this example, we extend Example 2 by introducing another blocking factor, transforming the design into an LSD.

Step 1: Creating the LSD

Generate the templates to determine the correct order of inputs.

designLSD(
  treatments = c(2, 2),
  squares = 4,
  reuse = "both",
  template = TRUE
)

Either beta or means can be provided, as in Example 2. Note that although the variance component template sets the order for row and col variances in vcomp, these values don’t impact power as long as sigma2 is fixed because treatment effects are tested against residual error. However, in designs where the error terms for treatment factors include variance components beyond residual error (like the next example), variance components can affect power for main plot factors. It's also recommended to use variance components as a guide to make an informed estimate of sigma2.

Create the LSD

lsd <- designLSD(
  treatments = c(2, 2),
  label = list(temp = c("T1", "T2"), dosage = c("D1", "D2")),
  squares = 4,
  reuse = "both",
  means = c(35, 40, 38, 41),
  vcomp = c(11, 2),
  sigma2 = 2
)

Once the design has been created, pwr.anova and pwr.contrast can be used to evaluate statistical power as demonstrated above.

Example 4: Split-plot Design

In this example, we create an SPD with two treatment factors, one at the main plot level and another at the subplot level. The design parameters are as follows:

  1. Treatments:

    • Main plot factor with 2 levels

    • Subplot factor with 3 levels

  2. Replicates:

    • Each main plot treatment has 5 plots, for a total of 10 plots.

    • This is a standard SPD, where the plots are the blocks at the subplot level and the subplot design follows an RCBD structure.

  3. Expected cell means are: $$ \begin{array}{c|c|c} & trt.sub1 & trt.sub2 & trt.sub3 \ \hline trt.main1 & 20 & 22 & 24\ trt.main2 & 22 & 24 & 28 \ \end{array} $$

  4. Variance Components:

    • Main-plot error: 4

    • Subplot (residual) error: 11

    • Total variance: 15

Generate input templates

designSPD(
  trt.main = 2,
  trt.sub = 3, 
  replicates = 10, 
  template = TRUE
)

Create the SPD

spd <- designSPD(
  trt.main = 2,
  trt.sub = 3, 
  replicates = 10, 
  means = c(20, 22, 22, 24, 24, 28),
  vcomp = 4,
  sigma2 = 11
)

One can verify that different values of main-plot error (vcomp) affect the power of the main plot factor, unlike in Example 3, where variance components did not influence power.

Example 5: Repeated measures

This example illustrates a repeated measures design, where three treatments (CON, TRT1, TRT2) are measured hourly over 8 hours. Within-subject correlations are modeled using an AR(1) structure with $\rho = 0.6$ and $\sigma^2 = 2$.

Design Details

1. Subjects: 6 per treatment group (total: 18 subjects).\ 2. Treatments: CON, TRT1, and TRT2.\ 3. Time Points: 8 hourly measurements.

4. Means:$$ \begin{array}{c|cccccccc} \textbf{Treatment} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} \ \hline \text{CON} & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \ \text{TRT1} & 2.50 & 3.50 & 3.98 & 4.03 & 3.68 & 3.35 & 3.02 & 2.94 \ \text{TRT2} & 3.50 & 4.54 & 5.80 & 5.84 & 5.49 & 4.71 & 4.08 & 3.78 \ \end{array} $$

Step 1: Creating the design

Create a data frame for the design

n_subject = 6 # Subjects per treatment
n_trt = 3 # Number of treatments
n_hour = 8 # Number of repeated measures (time points)
trt = c("CON", "TRT1", "TRT2")

df.rep <- data.frame(
  subject = as.factor(rep(seq_len(n_trt*n_subject), each = n_hour)),
  hour = as.factor(rep(seq_len(n_hour), n_subject*n_trt)),
  trt = rep(trt, each = n_subject*n_hour)
)

Input templates

Either values of beta or means are required in the following order:

mkdesign(formula = ~ trt*hour, data = df.rep)

Create the design

design.rep <- mkdesign(
formula = ~ trt*hour,
data = df.rep,
means =  c(1, 2.50, 3.5,
           1, 3.50, 4.54,
           1, 3.98, 5.80,
           1, 4.03, 5.4,
           1, 3.68, 5.49,
           1, 3.35, 4.71,
           1, 3.02, 4.08,
           1, 2.94, 3.78),
sigma2 = 2,
correlation = corAR1(value = 0.6, form = ~ hour|subject)
)

Step 2: Power calculation

Statistical power for main effects of treatment and time, and their interaction:

pwr.anova(design.rep)

Statistical power for treatment differences at each time point:

pwr.contrast(design.rep, which = "trt", by = "hour", contrast = "trt.vs.ctrl", p.adj = TRUE)[1:2]

As shown in Example 5, any design can be created using the mkdesign function as long as the appropriate data frame is provided. However, the package currently does not support non-normal response variables.

Fundamental concepts

pwr4exp is developed based on LMM theory. The general form of an LMM can be expressed as:

$$ y = X\beta + Zu + \varepsilon $$

where: $y$ represents the observations of the response variable, $\beta$ represents the fixed effect coefficients, $u$ denotes the random effects, where $u \sim N_q(0, G)$, $\varepsilon$ represents the random errors, where $\varepsilon \sim N_n(0, R)$, $X_{(n \times p)}$ and $Z_{(n \times q)}$ are the design matrices for the fixed and random effects, respectively.

It is assumed that $u$ and $\varepsilon$ are independent, and the marginal distribution of $y$ follows a normal distribution $y \sim N_n(X\beta, V)$, where:

$$ V = ZGZ^T + R $$

Inference on Treatment Effects

Inference on treatment effects often involves testing omnibus hypotheses and contrasts. These can be formulated using the general linear hypothesis:

$$ H_0: K\beta = 0 $$

where $K$ is a contrast matrix. When the variance-covariance parameters in $G$ and $R$ are known, the estimate of $\beta$ is:

$$ \hat{\beta} = (X^TV^{-1}X)^{-1}X^TV^{-1}y $$

And its variance is:

$$ C = (X^TV^{-1}X)^{-1} $$

The sampling distribution of $K'\hat{\beta}$ is:

$$ K'\hat{\beta} \sim N(0, K'CK) $$

However, in practical situations, the matrices $G$ and $R$ are unknown and must be estimated using methods like Maximum Likelihood (ML) or Restricted ML (REML). The estimate of $\beta$ is obtained by plugging in the estimated covariance matrices $\hat{V}$, where:

$$ \hat{V} = Z\hat{G}Z^T + \hat{R} $$

The resulting estimate of $\beta$ is:

$$ \hat{\beta} = (X^T\hat{V}^{-1}X)^{-1}X^T\hat{V}^{-1}y $$

And its estimated variance is:

$$ \hat{C} = (X^T\hat{V}^{-1}X)^{-1} $$

When testing the null hypothesis $H_0: K\beta = 0$, an approximate F-statistic is used. The F-statistic is given by:

$$ F = \frac{(K\hat{\beta})^T [K\hat{C}K^T]^{-1} (K\hat{\beta})}{v_1} $$

$F$ follows an approximate F-distribution $F(v_1, v_2)$ under $H_0$, where $v_1 = \text{rank}(K) \geq 1$ represents the numerator degrees of freedom (df), $v_2$ is the denominator df.

When $\text{rank}(K) = 1$, the F-statistic simplifies to the square of the t-statistic:

$$ F = t^2 $$where $t = \frac{k'\hat{\beta}}{\sqrt{k'\hat{C}K}}$ with $v_2$ df.

In balanced designs, where data is analyzed using a variance components model, commonly applied in experimental animal research, $v_2$​ can be precisely determined through degrees of freedom decomposition, as applied in analysis of variance (ANOVA).

However, for more general cases, $v_2$ must be approximated using methods.

The Satterthwaite approximation (Satterthwaite, 1946) for DF in t-tests can be calculated as outlined by Giesbrecht and Burns (1985):

$$ v_2 = \frac{2(k^T \hat{C} k)^2}{g^T A g} $$

where: $g$ is the gradient of $k^T C(\hat{\theta}) k$ with respect to $\theta$, the variance-covariance parameters in $V$, evaluated at $\hat{\theta}$. Matrix $A$ is the asymptotic variance-covariance matrix of $\hat{\theta}$, obtained from the information matrix of ML or REML estimation of $\hat{\theta}$ (Stroup, 2012).

In F-tests, $v_2$ can be calculated by following the procedures described by Fai and Cornelius (1996). First, $KC\hat{K}^T$ is decomposed to yield $KC\hat{K}^T = P^T D P$, where $P$ is an orthogonal matrix of eigen vectors, and $D$ is a diagonal matrix of eigenvalues. Define $k_m Ck_m^T$, where $k_m$ is the $m$-th row of $P K$, and let:

$$ v_m = \frac{2(D_m)^2}{g_m^T A g_m} $$

where: $D_m$is the $m$th diagonal element of $D$, $g_m$ is the gradient of $k_m C k_m^T$ with respect to $\theta$, evaluated at $\hat{\theta}$.

Then let:

$$ E = \sum_{m=1}^{v_1} \frac{v_m}{v_m - 2} I(v_m > 2) $$

where $I(v_m > 2)$ denotes the indicator function. The denominator DF $v_2$ is calculated as:

$$ v_2 = \frac{2E}{E - v_1} $$ The Satterthwaite approximation can be applied in power analysis by plugging in the assumed values of variance parameters (Stroup, 2002).

Power Calculation Under the Alternative Hypothesis

Under the alternative hypothesis $H_A: K'\beta \neq 0$, the F-statistic follows a non-central distribution $F(v_1, v_2, \phi)$, where $\phi$ is the non-centrality parameter that measures the departure from the null hypothesis $H_0$. The non-centrality parameter $\phi$ is given by:

$$ \phi = (K\hat{\beta})^T [K\hat{C}K^T]^{-1} (K\hat{\beta}) $$

Once the distribution of the F-statistic under $H_A$ is known, the power of the test can be calculated as the conditional probability of rejecting $H_0$ when $H_A$ is true:

$$ \text{Power} = P(\text{reject } H_0: F > F_{\text{crit}} \mid H_A) $$

Where: $F_{\text{crit}}$ is the critical value of the F-statistic used to reject $H_0$, determined such that $P(F > F_{\text{crit}} \mid H_0) = \alpha$, where $\alpha$ is the type I error rate.

The determination of the degrees of freedom $v_1$ and $v_2$, as well as the non-centrality parameter $\phi$, are critical steps for power calculation.

Generally, power analysis requires specifying the following components:

A key aspect of conducting a valid power analysis is obtaining reasonable estimates for the magnitude of the parameters that will be used in the model. This includes:

Performing a power analysis with unrealistic parameter magnitudes can lead to incorrect conclusions, either overestimating the likelihood of detecting a treatment effect or requiring an unnecessarily large sample size.

References

Fai, A. H., & Cornelius, P. L. (1996). Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments. Journal of Statistical Computation and Simulation, 54(4), 363–378.

Giesbrecht, F. G., & Burns, J. C. (1985). Two-stage analysis based on a mixed model: Large-sample asymptotic theory and small-sample simulation results. Biometrics, 41(2), 477–486.

Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6), 110–114.

Stroup, W. W. (2002). Power analysis based on spatial effects mixed models: A tool for comparing design and analysis strategies in the presence of spatial variability. Journal of Agricultural, Biological, and Environmental Statistics, 7(4), 491–511.

Stroup, W. W. (2012). Generalized linear mixed models: Modern concepts, methods, and applications (1st ed.). CRC Press.



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